

<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
  <meta charset="utf-8">
  
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="Description" content="scikit-learn: machine learning in Python">

  
  <title>Spectral clustering for image segmentation &mdash; scikit-learn 0.22 documentation</title>
  
  <link rel="canonical" href="http://scikit-learn.org/stable/auto_examples/cluster/plot_segmentation_toy.html" />

  
  <link rel="shortcut icon" href="../../_static/favicon.ico"/>
  

  <link rel="stylesheet" href="../../_static/css/vendor/bootstrap.min.css" type="text/css" />
  <link rel="stylesheet" href="../../_static/gallery.css" type="text/css" />
  <link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../../" src="../../_static/documentation_options.js"></script>
<script src="../../_static/jquery.js"></script> 
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
  <div class="container-fluid sk-docs-container px-0">
      <a class="navbar-brand py-0" href="../../index.html">
        <img
          class="sk-brand-img"
          src="../../_static/scikit-learn-logo-small.png"
          alt="logo"/>
      </a>
    <button
      id="sk-navbar-toggler"
      class="navbar-toggler"
      type="button"
      data-toggle="collapse"
      data-target="#navbarSupportedContent"
      aria-controls="navbarSupportedContent"
      aria-expanded="false"
      aria-label="Toggle navigation"
    >
      <span class="navbar-toggler-icon"></span>
    </button>

    <div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
      <ul class="navbar-nav mr-auto">
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../install.html">Install</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../user_guide.html">User Guide</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../modules/classes.html">API</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../index.html">Examples</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../getting_started.html">Getting Started</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../tutorial/index.html">Tutorial</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../glossary.html">Glossary</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../developers/index.html">Development</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../faq.html">FAQ</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../related_projects.html">Related packages</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../roadmap.html">Roadmap</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../about.html">About us</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn">GitHub</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html">Other Versions</a>
        </li>
        <li class="nav-item dropdown nav-more-item-dropdown">
          <a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
          <div class="dropdown-menu" aria-labelledby="navbarDropdown">
              <a class="sk-nav-dropdown-item dropdown-item" href="../../getting_started.html">Getting Started</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../tutorial/index.html">Tutorial</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../glossary.html">Glossary</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../developers/index.html">Development</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../faq.html">FAQ</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../related_projects.html">Related packages</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../roadmap.html">Roadmap</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../about.html">About us</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn">GitHub</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html">Other Versions</a>
          </div>
        </li>
      </ul>
      <div id="searchbox" role="search">
          <div class="searchformwrapper">
          <form class="search" action="../../search.html" method="get">
            <input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
            <input class="sk-search-text-btn" type="submit" value="Go" />
          </form>
          </div>
      </div>
    </div>
  </div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
    <input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
    <label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
    <div id="sk-sidebar-wrapper" class="border-right">
      <div class="sk-sidebar-toc-wrapper">
        <div class="sk-sidebar-toc-logo">
          <a href="../../index.html">
            <img
              class="sk-brand-img"
              src="../../_static/scikit-learn-logo-small.png"
              alt="logo"/>
          </a>
        </div>
        <div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
            <a href="plot_cluster_iris.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="K-means Clustering">Prev</a><a href="../index.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Examples">Up</a>
            <a href="plot_coin_ward_segmentation.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="A demo of structured Ward hierarchical clustering on an image of coins">Next</a>
        </div>
        <div class="alert alert-danger p-1 mb-2" role="alert">
          <p class="text-center mb-0">
          <strong>scikit-learn 0.22</strong><br/>
          <a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
          </p>
        </div>
        <div class="alert alert-warning p-1 mb-2" role="alert">
          <p class="text-center mb-0">
            Please <a class="font-weight-bold" href="../../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
          </p>
        </div>
          <div class="sk-sidebar-toc">
            <ul>
<li><a class="reference internal" href="#">Spectral clustering for image segmentation</a></li>
</ul>

          </div>
      </div>
    </div>
    <div id="sk-page-content-wrapper">
      <div class="sk-page-content container-fluid body px-md-3" role="main">
        
  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-cluster-plot-segmentation-toy-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
</div>
<div class="sphx-glr-example-title section" id="spectral-clustering-for-image-segmentation">
<span id="sphx-glr-auto-examples-cluster-plot-segmentation-toy-py"></span><h1>Spectral clustering for image segmentation<a class="headerlink" href="#spectral-clustering-for-image-segmentation" title="Permalink to this headline">¶</a></h1>
<p>In this example, an image with connected circles is generated and
spectral clustering is used to separate the circles.</p>
<p>In these settings, the <a class="reference internal" href="../../modules/clustering.html#spectral-clustering"><span class="std std-ref">Spectral clustering</span></a> approach solves the problem
know as ‘normalized graph cuts’: the image is seen as a graph of
connected voxels, and the spectral clustering algorithm amounts to
choosing graph cuts defining regions while minimizing the ratio of the
gradient along the cut, and the volume of the region.</p>
<p>As the algorithm tries to balance the volume (ie balance the region
sizes), if we take circles with different sizes, the segmentation fails.</p>
<p>In addition, as there is no useful information in the intensity of the image,
or its gradient, we choose to perform the spectral clustering on a graph
that is only weakly informed by the gradient. This is close to performing
a Voronoi partition of the graph.</p>
<p>In addition, we use the mask of the objects to restrict the graph to the
outline of the objects. In this example, we are interested in
separating the objects one from the other, and not from the background.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Authors:  Emmanuelle Gouillart &lt;emmanuelle.gouillart@normalesup.org&gt;</span>
<span class="c1">#           Gael Varoquaux &lt;gael.varoquaux@normalesup.org&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>

<span class="kn">from</span> <span class="nn">sklearn.feature_extraction</span> <span class="kn">import</span> <span class="n">image</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">spectral_clustering</span>

<span class="n">l</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">indices</span><span class="p">((</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="p">))</span>

<span class="n">center1</span> <span class="o">=</span> <span class="p">(</span><span class="mi">28</span><span class="p">,</span> <span class="mi">24</span><span class="p">)</span>
<span class="n">center2</span> <span class="o">=</span> <span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
<span class="n">center3</span> <span class="o">=</span> <span class="p">(</span><span class="mi">67</span><span class="p">,</span> <span class="mi">58</span><span class="p">)</span>
<span class="n">center4</span> <span class="o">=</span> <span class="p">(</span><span class="mi">24</span><span class="p">,</span> <span class="mi">70</span><span class="p">)</span>

<span class="n">radius1</span><span class="p">,</span> <span class="n">radius2</span><span class="p">,</span> <span class="n">radius3</span><span class="p">,</span> <span class="n">radius4</span> <span class="o">=</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">14</span>

<span class="n">circle1</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center1</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center1</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">&lt;</span> <span class="n">radius1</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">circle2</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center2</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center2</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">&lt;</span> <span class="n">radius2</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">circle3</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center3</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center3</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">&lt;</span> <span class="n">radius3</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">circle4</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center4</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center4</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">&lt;</span> <span class="n">radius4</span> <span class="o">**</span> <span class="mi">2</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># 4 circles</span>
<span class="n">img</span> <span class="o">=</span> <span class="n">circle1</span> <span class="o">+</span> <span class="n">circle2</span> <span class="o">+</span> <span class="n">circle3</span> <span class="o">+</span> <span class="n">circle4</span>

<span class="c1"># We use a mask that limits to the foreground: the problem that we are</span>
<span class="c1"># interested in here is not separating the objects from the background,</span>
<span class="c1"># but separating them one from the other.</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span>

<span class="n">img</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">img</span> <span class="o">+=</span> <span class="mi">1</span> <span class="o">+</span> <span class="mf">0.2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">img</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>

<span class="c1"># Convert the image into a graph with the value of the gradient on the</span>
<span class="c1"># edges.</span>
<span class="n">graph</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">img_to_graph</span><span class="p">(</span><span class="n">img</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>

<span class="c1"># Take a decreasing function of the gradient: we take it weakly</span>
<span class="c1"># dependent from the gradient the segmentation is close to a voronoi</span>
<span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">/</span> <span class="n">graph</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>

<span class="c1"># Force the solver to be arpack, since amg is numerically</span>
<span class="c1"># unstable on this example</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">spectral_clustering</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">n_clusters</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">eigen_solver</span><span class="o">=</span><span class="s1">&#39;arpack&#39;</span><span class="p">)</span>
<span class="n">label_im</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">mask</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">)</span>
<span class="n">label_im</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="n">labels</span>

<span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">img</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">label_im</span><span class="p">)</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># 2 circles</span>
<span class="n">img</span> <span class="o">=</span> <span class="n">circle1</span> <span class="o">+</span> <span class="n">circle2</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span>
<span class="n">img</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>

<span class="n">img</span> <span class="o">+=</span> <span class="mi">1</span> <span class="o">+</span> <span class="mf">0.2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="o">*</span><span class="n">img</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>

<span class="n">graph</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">img_to_graph</span><span class="p">(</span><span class="n">img</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
<span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">/</span> <span class="n">graph</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>

<span class="n">labels</span> <span class="o">=</span> <span class="n">spectral_clustering</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">n_clusters</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">eigen_solver</span><span class="o">=</span><span class="s1">&#39;arpack&#39;</span><span class="p">)</span>
<span class="n">label_im</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">mask</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">)</span>
<span class="n">label_im</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="n">labels</span>

<span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">img</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">label_im</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes  0.000 seconds)</p>
<div class="sphx-glr-footer class sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-cluster-plot-segmentation-toy-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.22.X?urlpath=lab/tree/notebooks/auto_examples/cluster/plot_segmentation_toy.ipynb"><img alt="https://mybinder.org/badge_logo.svg" src="https://mybinder.org/badge_logo.svg" width="150px" /></a>
</div>
<div class="sphx-glr-download docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/c5f78418d947efdfcd8779e8974e3b03/plot_segmentation_toy.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_segmentation_toy.py</span></code></a></p>
</div>
<div class="sphx-glr-download docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/2b2ceafc276c3d847814de985fe3858d/plot_segmentation_toy.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_segmentation_toy.ipynb</span></code></a></p>
</div>
</div>
<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</div>


      </div>
    <div class="container">
      <footer class="sk-content-footer">
            &copy; 2007 - 2019, scikit-learn developers (BSD License).
          <a href="../../_sources/auto_examples/cluster/plot_segmentation_toy.rst.txt" rel="nofollow">Show this page source</a>
      </footer>
    </div>
  </div>
</div>
<script src="../../_static/js/vendor/bootstrap.min.js"></script>

<script>
    window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
    ga('create', 'UA-22606712-2', 'auto');
    ga('set', 'anonymizeIp', true);
    ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>


<script>
$(document).ready(function() {
    /* Add a [>>>] button on the top-right corner of code samples to hide
     * the >>> and ... prompts and the output and thus make the code
     * copyable. */
    var div = $('.highlight-python .highlight,' +
                '.highlight-python3 .highlight,' +
                '.highlight-pycon .highlight,' +
		'.highlight-default .highlight')
    var pre = div.find('pre');

    // get the styles from the current theme
    pre.parent().parent().css('position', 'relative');
    var hide_text = 'Hide prompts and outputs';
    var show_text = 'Show prompts and outputs';

    // create and add the button to all the code blocks that contain >>>
    div.each(function(index) {
        var jthis = $(this);
        if (jthis.find('.gp').length > 0) {
            var button = $('<span class="copybutton">&gt;&gt;&gt;</span>');
            button.attr('title', hide_text);
            button.data('hidden', 'false');
            jthis.prepend(button);
        }
        // tracebacks (.gt) contain bare text elements that need to be
        // wrapped in a span to work with .nextUntil() (see later)
        jthis.find('pre:has(.gt)').contents().filter(function() {
            return ((this.nodeType == 3) && (this.data.trim().length > 0));
        }).wrap('<span>');
    });

    // define the behavior of the button when it's clicked
    $('.copybutton').click(function(e){
        e.preventDefault();
        var button = $(this);
        if (button.data('hidden') === 'false') {
            // hide the code output
            button.parent().find('.go, .gp, .gt').hide();
            button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'hidden');
            button.css('text-decoration', 'line-through');
            button.attr('title', show_text);
            button.data('hidden', 'true');
        } else {
            // show the code output
            button.parent().find('.go, .gp, .gt').show();
            button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'visible');
            button.css('text-decoration', 'none');
            button.attr('title', hide_text);
            button.data('hidden', 'false');
        }
    });

	/*** Add permalink buttons next to glossary terms ***/
	$('dl.glossary > dt[id]').append(function() {
		return ('<a class="headerlink" href="#' +
			    this.getAttribute('id') +
			    '" title="Permalink to this term">¶</a>');
	});
  /*** Hide navbar when scrolling down ***/
  // Returns true when headerlink target matches hash in url
  (function() {
    hashTargetOnTop = function() {
        var hash = window.location.hash;
        if ( hash.length < 2 ) { return false; }

        var target = document.getElementById( hash.slice(1) );
        if ( target === null ) { return false; }

        var top = target.getBoundingClientRect().top;
        return (top < 2) && (top > -2);
    };

    // Hide navbar on load if hash target is on top
    var navBar = document.getElementById("navbar");
    var navBarToggler = document.getElementById("sk-navbar-toggler");
    var navBarHeightHidden = "-" + navBar.getBoundingClientRect().height + "px";
    var $window = $(window);

    hideNavBar = function() {
        navBar.style.top = navBarHeightHidden;
    };

    showNavBar = function() {
        navBar.style.top = "0";
    }

    if (hashTargetOnTop()) {
        hideNavBar()
    }

    var prevScrollpos = window.pageYOffset;
    hideOnScroll = function(lastScrollTop) {
        if (($window.width() < 768) && (navBarToggler.getAttribute("aria-expanded") === 'true')) {
            return;
        }
        if (lastScrollTop > 2 && (prevScrollpos <= lastScrollTop) || hashTargetOnTop()){
            hideNavBar()
        } else {
            showNavBar()
        }
        prevScrollpos = lastScrollTop;
    };

    /*** high preformance scroll event listener***/
    var raf = window.requestAnimationFrame ||
        window.webkitRequestAnimationFrame ||
        window.mozRequestAnimationFrame ||
        window.msRequestAnimationFrame ||
        window.oRequestAnimationFrame;
    var lastScrollTop = $window.scrollTop();

    if (raf) {
        loop();
    }

    function loop() {
        var scrollTop = $window.scrollTop();
        if (lastScrollTop === scrollTop) {
            raf(loop);
            return;
        } else {
            lastScrollTop = scrollTop;
            hideOnScroll(lastScrollTop);
            raf(loop);
        }
    }
  })();
});

</script>
    
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
    
</body>
</html>